*SINUM*

**Inserted:** 3 dec 2013

**Last Updated:** 27 sep 2015

**Year:** 2013

**Abstract:**

We consider the multiphase shape optimization problem

$\min\Big\{\sum_{i=1}^h\big(\lambda_1(\Omega_i)+c \vert

\Omega_i \vert

\big):\ \Omega_i\ \hbox{open},\ \Omega_i\subset D,\ \Omega_i\cap\Omega_j=\emptyset \Big\},$

where $c>0$ is a given constant and $D\subset\mathbb{R}^2$ is a bounded open set with Lipschitz boundary. We give some new results concerning the qualitative properties of the optimal sets and the regularity of the corresponding eigenfunctions. We also provide some numerical results for the optimal configuration.

**Keywords:**
monotonicity formula, shape optimization, eigenvalues, multiphase, optimal partition

**Download:**